Radiation and scattering of waves on an elastic half-space; a non-commutative matrix Wiener-Hopf problem

Many problems in linear elastodynamics, or dynamic fracture mechanics, can be reduced to Wiener-Hopf functional equations defined in some region of a complex transform plane. The key step in the solution of a Wiener-Hopf equation, which is to decompose the kernel into a product of two factors with particular analyticity properties, can be accomplished explicitly for scalar kernels. However, apart from special matrices which yield commutative factorizations, no procedure has yet been devised to exactly factorize general matrix kernels. It is the aim of this article to show that a new procedure for obtaining approximate factors of matrix kernels is applicable to the class of matrix kernels found in elasticity. This is performed, for ease of exposition, by way of a simple but non-trivial example. Via the substitution of a scalar function in the kernel by its Pade approximant, an approximate solution to the boundary value problem is obtained explicitly for three different forcing cases. The approximate factorization technique described herein is simple to apply, is shown to converge to the exact result as the Pade number increases, and a maximum error bound is easily obtained. Numerical evaluation of the explicit results reveals that convergence to the exact solution is extremely rapid, which is confirmed by a global energy balance calculation.